3.2406 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=171 \[ -\frac{(5 x+3)^{3/2} (1-2 x)^{5/2}}{9 (3 x+2)^3}+\frac{115 (5 x+3)^{3/2} (1-2 x)^{3/2}}{108 (3 x+2)^2}+\frac{365 (5 x+3)^{3/2} \sqrt{1-2 x}}{216 (3 x+2)}-\frac{845}{648} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{362}{243} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{215 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1944 \sqrt{7}} \]

[Out]

(-845*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/648 - ((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(9*(2 + 3*x)^3) + (115*(1 - 2*x)^(3
/2)*(3 + 5*x)^(3/2))/(108*(2 + 3*x)^2) + (365*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(216*(2 + 3*x)) + (362*Sqrt[10]*A
rcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/243 + (215*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1944*Sqrt[7])

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Rubi [A]  time = 0.0653066, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {97, 149, 154, 157, 54, 216, 93, 204} \[ -\frac{(5 x+3)^{3/2} (1-2 x)^{5/2}}{9 (3 x+2)^3}+\frac{115 (5 x+3)^{3/2} (1-2 x)^{3/2}}{108 (3 x+2)^2}+\frac{365 (5 x+3)^{3/2} \sqrt{1-2 x}}{216 (3 x+2)}-\frac{845}{648} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{362}{243} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{215 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1944 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^4,x]

[Out]

(-845*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/648 - ((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(9*(2 + 3*x)^3) + (115*(1 - 2*x)^(3
/2)*(3 + 5*x)^(3/2))/(108*(2 + 3*x)^2) + (365*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(216*(2 + 3*x)) + (362*Sqrt[10]*A
rcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/243 + (215*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1944*Sqrt[7])

Rule 97

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])

Rule 149

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

Rule 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

Rule 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx &=-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac{1}{9} \int \frac{\left (-\frac{15}{2}-40 x\right ) (1-2 x)^{3/2} \sqrt{3+5 x}}{(2+3 x)^3} \, dx\\ &=-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac{115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{108 (2+3 x)^2}-\frac{1}{54} \int \frac{\left (-\frac{2325}{4}-735 x\right ) \sqrt{1-2 x} \sqrt{3+5 x}}{(2+3 x)^2} \, dx\\ &=-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac{115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{108 (2+3 x)^2}+\frac{365 \sqrt{1-2 x} (3+5 x)^{3/2}}{216 (2+3 x)}+\frac{1}{162} \int \frac{\sqrt{3+5 x} \left (\frac{6975}{8}+\frac{2535 x}{2}\right )}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=-\frac{845}{648} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac{115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{108 (2+3 x)^2}+\frac{365 \sqrt{1-2 x} (3+5 x)^{3/2}}{216 (2+3 x)}-\frac{1}{972} \int \frac{-\frac{57705}{4}-21720 x}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{845}{648} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac{115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{108 (2+3 x)^2}+\frac{365 \sqrt{1-2 x} (3+5 x)^{3/2}}{216 (2+3 x)}-\frac{215 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{3888}+\frac{1810}{243} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{845}{648} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac{115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{108 (2+3 x)^2}+\frac{365 \sqrt{1-2 x} (3+5 x)^{3/2}}{216 (2+3 x)}-\frac{215 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{1944}+\frac{1}{243} \left (724 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )\\ &=-\frac{845}{648} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac{115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{108 (2+3 x)^2}+\frac{365 \sqrt{1-2 x} (3+5 x)^{3/2}}{216 (2+3 x)}+\frac{362}{243} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )+\frac{215 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{1944 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.167385, size = 131, normalized size = 0.77 \[ \frac{-21 \sqrt{5 x+3} \left (8640 x^4+64362 x^3+38127 x^2-15626 x-10304\right )-20272 \sqrt{10-20 x} (3 x+2)^3 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )+215 \sqrt{7-14 x} (3 x+2)^3 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{13608 \sqrt{1-2 x} (3 x+2)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^4,x]

[Out]

(-21*Sqrt[3 + 5*x]*(-10304 - 15626*x + 38127*x^2 + 64362*x^3 + 8640*x^4) - 20272*Sqrt[10 - 20*x]*(2 + 3*x)^3*A
rcSin[Sqrt[5/11]*Sqrt[1 - 2*x]] + 215*Sqrt[7 - 14*x]*(2 + 3*x)^3*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]
)/(13608*Sqrt[1 - 2*x]*(2 + 3*x)^3)

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Maple [B]  time = 0.011, size = 270, normalized size = 1.6 \begin{align*} -{\frac{1}{27216\, \left ( 2+3\,x \right ) ^{3}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 5805\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-547344\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{3}+11610\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-1094688\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-181440\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+7740\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-729792\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-1442322\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+1720\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -162176\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -1521828\,x\sqrt{-10\,{x}^{2}-x+3}-432768\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^4,x)

[Out]

-1/27216*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(5805*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3-5473
44*10^(1/2)*arcsin(20/11*x+1/11)*x^3+11610*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-1094
688*10^(1/2)*arcsin(20/11*x+1/11)*x^2-181440*x^3*(-10*x^2-x+3)^(1/2)+7740*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2
)/(-10*x^2-x+3)^(1/2))*x-729792*10^(1/2)*arcsin(20/11*x+1/11)*x-1442322*x^2*(-10*x^2-x+3)^(1/2)+1720*7^(1/2)*a
rctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-162176*10^(1/2)*arcsin(20/11*x+1/11)-1521828*x*(-10*x^2-x+3)
^(1/2)-432768*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^3

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Maxima [A]  time = 2.54698, size = 217, normalized size = 1.27 \begin{align*} \frac{125}{378} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{3 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{25 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{84 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{1825}{756} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{181}{243} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{215}{27216} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{655}{4536} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{65 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{504 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^4,x, algorithm="maxima")

[Out]

125/378*(-10*x^2 - x + 3)^(3/2) + 1/3*(-10*x^2 - x + 3)^(5/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 25/84*(-10*x^2 -
x + 3)^(5/2)/(9*x^2 + 12*x + 4) + 1825/756*sqrt(-10*x^2 - x + 3)*x + 181/243*sqrt(10)*arcsin(20/11*x + 1/11) -
215/27216*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 655/4536*sqrt(-10*x^2 - x + 3) - 65/504
*(-10*x^2 - x + 3)^(3/2)/(3*x + 2)

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Fricas [A]  time = 1.62804, size = 486, normalized size = 2.84 \begin{align*} \frac{215 \, \sqrt{7}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{14 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) - 20272 \, \sqrt{10}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) + 42 \,{\left (4320 \, x^{3} + 34341 \, x^{2} + 36234 \, x + 10304\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{27216 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^4,x, algorithm="fricas")

[Out]

1/27216*(215*sqrt(7)*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)
/(10*x^2 + x - 3)) - 20272*sqrt(10)*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)
*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 42*(4320*x^3 + 34341*x^2 + 36234*x + 10304)*sqrt(5*x + 3)*sqrt(-2*x + 1))/
(27*x^3 + 54*x^2 + 36*x + 8)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**(5/2)*(3+5*x)**(3/2)/(2+3*x)**4,x)

[Out]

Timed out

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Giac [B]  time = 3.79466, size = 545, normalized size = 3.19 \begin{align*} -\frac{43}{54432} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{181}{243} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{4}{81} \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{11 \,{\left (67 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 56000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 65464000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{108 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^4,x, algorithm="giac")

[Out]

-43/54432*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))
^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) + 181/243*sqrt(10)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)*
((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) + 4/81*sqrt(5)*s
qrt(5*x + 3)*sqrt(-10*x + 5) - 11/108*(67*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqr
t(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5 + 56000*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt
(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 - 65464000*sqrt(10)*((sqrt(2)*sqrt(-10*x +
5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))))/(((sqrt(2)*sqrt(-10*x +
5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^3